Low-rank tensor methods for linear systems and eigenvalue problems

This thesis is concerned with methods for the approximate solution of high-dimensional linear systems and eigenvalue problems, using low-rank tensor techniques. In recent years, new low-rank tensor decompositions have been developed, which allow the black box approximation of a tensor at a given accuracy, and still have storage requirements that grow only linearly in the number of dimensions for a fixed rank. We have implemented a Matlab toolbox allowing the storage of a tensor in the Hierarchical Tucker decomposition (HTD), one of the aforementioned low-rank tensor decompositions. Additionally, basic operations such as the addition or inner product of two tensors in HTD are featured. Moreover, we introduce a new variant of approximation in HTD. A detailed description of all operations in the htucker toolbox is provided, as well as many examples. Applications to parameter-dependent linear systems, stochastic elliptic partial differential equations (PDEs), high-dimensional parabolic PDEs and high-dimensional elliptic PDE eigenvalue problems are presented. Through discretization on a tensor grid, our examples result in linear systems, A(X ) = B, and eigenvalue problems, A(X ) = λX . The operator A : Rn1×···×nd → Rn1×···×nd typically is a short sum of Kronecker products of matrices. The approximability of the solution tensor in HTD is discussed for the case of analytically parameter-dependent linear systems. For the case of linear systems, we adapt well-known vector iterations, specifically the preconditioned Richardson, the preconditioned conjugate gradients and the preconditioned BiCGstab methods, to the low-rank case by replacing the iterated tensors with a low-rank approximation in each step. Different schemes for choosing the accuracy of these approximations in each step are presented. The application to a certain space-time discretization of parabolic evolution equations is discussed in detail. For high-dimensional eigenvalue problems a variant of the LOBPCG method, a vector iteration for determining the smallest eigenvalue and the respective eigenvector of a matrix, is developed, following the ideas used for high-dimensional linear systems. Alternatively, we explore the DMRG or (M)ALS method, first introduced in the context of computational quantum chemistry. Finally, a combination of both (M)ALS and our variant of LOBPCG is presented.